Proof of Propositions 3, 4, and 5, and of Corollaries 3 and 4



 0 0 1

1

2 1

2 0

a₃₁ a₃₂ a₃₃





Note that this is also submatrix a, excluded in Proposition 2.

Altogether we obtain for a 6=

µ 0 0

1 2

1 2

¶

∧ a 6=

µ ₁

2 1

0 02

¶

min

α∈{D₃^x²,D^o₃,D³₃,D₃^{T ²}}

{π₃(α)} = min

ρ non-symmetric{π₃} > 1

4 (72)

7.1.4 Proof of Corollary 2 W.l.o.g we assume a =

µ 0 0

1

2 1

2

¶

=⇒ Player 3 must offer Player i at least a_3i = µ_i and Player j at least a_3j = x_j+ ² (i, j = 1, 2 i 6= j) to prevent D₂ from being a single-proposal equilibrium with π₃ = 0. =⇒ D₃ = (¹₄,¹₂ + ²,¹₄ − ²) or D₃ = (¹₂ + ²,¹₄,¹₄ − ²) with the correlated equilibrium C₂₃ is the best reaction for Player 3, resulting in the payoffs π₁ = π₂ = ₁₆⁷ + ^²₄ and π₃ = ¹₈ − ₂^² (given D₁ and D₂, this is also the worst situation for Player 3).

7.2 Proof of Propositions 3, 4, and 5, and of Corollaries 3 and 4

We prove these propositions and corollaries by constructing D₂ in such a way that we satisfy different constraint sets in the proofs of Propositions 1 and 2. But since we have only shown Propositions 1 and 2 for representative matrices a, we sometimes have to interchange indices in the constraint sets to adapt them for the following proofs.

7.2.1 Proof of Proposition 3 single-proposal equilibrium =⇒ This requires the following rank matrix:

A =

This can be satisfied if constraint set (i) of D^x₃ holds. The feasible set SA1 is then determined by²²

1. 2a21 ≥ a11 =⇒ a11 ≤ 1 2. 2a₁₂ < a₂₂ =⇒ a₁₂ < ¹₄ 3. µ₁ > x =⇒ a₁₁ > ¹₂

4. µ2+ 2y1− x1 ≥ x =⇒ a12 ≥ 2a11−³₂ After comparison of the inequalities, only 3. and 4. remain.

=⇒ S_A1 = ©¡₁

Minimum payoff for Player 2:

π₂^min(A₁) = 1

22The bold numbers will denote those constraints which are binding.

After comparison of the inequalities, only 2. and 5. remain.

Minimum payoff for Player 2:

π₂^min(A₂) = 1

2 (78)

A₃) Suppose D₂ = (a₁₂, a₁₁− ², a₁₃+ ²) and D₃ = (0, µ₂, 1 − µ₂) is a single-proposal equilibrium =⇒ This requires the following rank matrix:

A = feasible set S_A3 is then determined by

1. 2a₂₁ < a₁₁ =⇒ a₁₁ > ³₄ 2. a₂₃ ≥ a₁₃ =⇒ ² > 0 3. 1 − µ₂+ a₁₃ > 2a₂₃ =⇒ a₁₂ ≥ ²⁰ − a₁₁ 4. µ₂ > ³₈ =⇒ a₁₂ > ³₄ − a₁₁ After comparison of the inequalities, constraints 1. and 4. remain.

=⇒ S_A3 = ©¡₃

4, 1¤

× [0, 1 − a₁₁]ª

(80) (See Figure 1)

Minimum payoff for Player 2:

π₂^min(A₃) > 3

8 (81)

A₄) To obtain complete cover for A there remains the line given by a₁₁= ¹₂. Suppose D2 = (¹₂− ²,¹₂− ², 2²) (² < ¹₂− a12) and D3 = (0,¹₂ − ²,¹₂ + ²) is the best reaction for Player 3 and a single-proposal equilibrium =⇒ This requires the following rank matrix:

D₁ = (0.5 0.4 0.1) D^A₂⁴ = (0.5 − ² 0.5 − ² 2²) D₃ = (0.00 0.5 − ² 0.5 + ²)

Taking a₁₁= ¹₂, a₂₁= a₂₂= ¹₂ − ², we see that either of the constraints on A₁ or A₂ hold and thus we obtain

=⇒ S_A4 = ©£₁

Minimum payoff for Player 2:

π₂^min(A₄) = 1 reaction for Player 3 and a single-proposal equilibrium =⇒ This requires the following rank matrix:

This can be satisfied if the constraint set (ii) of D^x₃ holds (by construction we have x = a12). The feasible set SB1 is then determined by After comparison of the inequalities, only 3., 6., and 7. remain.

=⇒ SB1 = ©£

(See Figure 2 )

Minimum payoff for Player 2:

π₂^min(B₁) = 1

2 (87)

B₂) Suppose D₂ = (³₄−a₁₁+3²,³₄−a₁₂+2², a₁₁+a₁₂−¹₂−5²) and D₃ = (0, µ₂, 1−µ₂) is the best reaction for Player 3 and a single-proposal equilibrium =⇒ This requires the following rank matrix:

A =

This can be satisfied if the constraints of D^µ₃ hold. The feasible set SB2 is then determined by

1. µ₂ < µ₁ =⇒ ² > 0 2. 2a₁₁ < a₂₁ =⇒ a₁₁ ≤ ¹₄ 3. a23 ≤ a13 =⇒ a12 ≤ ³₄ − a11

4. 1 − µ₂+ a₂₃ > 2a₁₃ =⇒ a₁₂ > ⁵₈ − a₁₁ After comparison of the inequalities, only 3. and 4. remain.

=⇒ SB2 = ©£

Minimum payoff for Player 2:

π₂^min(B₂) > 3

8 (90)

B3) Suppose D2 = (a12+ ², 1 − (a12 + ²), 0) and D3 = (0, a12, 1 − a12) is the best reaction for Player 3 and a single-proposal equilibrium =⇒ This requires the following rank matrix:

This can be satisfied if the constraint set (iv) of D₃^x holds. The feasible set S_B3 is then determined by

1. 2a₂₂ ≥ a₁₂ =⇒ a₁₂ < ²₃ 2. 2a11 ≥ a21 =⇒ a12 < 2a11

3. µ₁+ 2a₂₂− a₁₂ ≥ a₁₂ =⇒ a₁₂ < ¹₇a₁₁+⁴₇ 4. µ₂+ 2a₁₁− a₂₁ ≥ a₁₂ =⇒ a₁₂ < ¹₄ + a₁₁ After comparison of the inequalities only 4. remains.

=⇒ S_B3 = ©£₁

Minimum payoff for Player 2:

π₂^min(B₃) = 1

2 (93)

B₄) Suppose D₂ = (a₁₂ − ¹₄ − 2²,³₄ − a₁₂ + 2², 0) and D₃ = (0, µ₂, 1 − µ₂) is the best reaction for Player 3 and a single-proposal equilibrium =⇒ This requires the following rank matrix:

A =

This can be satisfied if the constraints of D^µ₃ hold (a₂₂ has to be non-negative).

The feasible set S_B4 is then determined by

1. µ₂ < µ₁ =⇒ a₁₂ > ¹₂ − a₁₁ 2. 2a₁₁ < a₂₁ =⇒ a₁₂ > 2a₁₁− ¹₄ 3. a23 ≤ a13 =⇒ a12 ≤ 1 − a11

4. 1 − µ₂+ a₂₃ > 2a₁₃ =⇒ a₁₂ > ¹¹₁₆ − a₁₁

5. a₂₂ ≥ 0 =⇒ a₁₂ ≤ ³₄

After comparison of the inequalities, only 2., 4. and 5. remain.

=⇒ S_B4 = ©£

Minimum payoff for Player 2:

π₂^min(B₄) > 3

8 (96)

B₅) Suppose D₂ = (⁵₈ + 4²,³₈ − 4², 0) and D₃ = (0, µ₂, 1 − µ₂) is the best reaction for Player 3 and implies the correlated equilibrium C₁₃ =⇒ This requires the following rank matrix:

This can be satisfied if the constraint set (ii) of D₃^cor holds (a₁₂= x). The feasible set S_B5 is then determined by the following and the payoff is greater than ³₈

1. 2a₂₂ ≥ a₁₂ =⇒ a₁₂ < ³₄

After comparison of the inequalities only 4. and 10. are remaining.

=⇒ SB5 = ©£

Minimum payoff for Player 2:

π₂^min(B₅) ≥ 51

96− ² (99)

B₆) Suppose D₂ = (a₁₂+ 4², 1 − (a₁₂+ 4²), 0) and D₃ = (0, µ₂, 1 − µ₂) is the best reaction for Player 3 and implies the correlated equilibrium C₁₃=⇒ This requires the following rank matrix:

A =

This can be satisfied if the constraint set (i) of D^cor₃ holds (by construction we have x = a₁₂). The feasible set S_B6 is then determined by

1. 2a₂₂ ≥ a₁₂ =⇒ a₁₂ < ²₃ 2. 2a₁₁ < a₂₁ =⇒ a₁₂ ≥ 2a₁₁

3. µ₂ < x =⇒ a₁₂ ≥ ¹₂

4. µ₁+ 2a₂₂− a₁₂ ≥ µ₂ =⇒ a₁₂ < ³₅ + ^a11₅ 5. a₂₃ < a₁₃ =⇒ a₁₂ < 1 − a₁₁ 6. 1 − µ₂ + a₂₃ ≤ 2a₁₃ =⇒ a₁₂ < ³₄ − a₁₁ 7. ¹₂(1 − µ₂+ a₁₃) > 1 − x =⇒ a₁₂ > ¹₂ + a₁₁ After comparison of the inequalities, only 4. and 7. remain.

=⇒ S_B6 = ©£

0,¹₈¤

×£₁

2 + a₁₁,³₅ + ¹₅a₁₁¢ª

(101) (See Figure 2)

Minimum payoff for Player 2:

π^min₂ (B₆) ≥ 1

2 − ² (102)

B7) Suppose D2 = (1, 0, 0) and D3 = (0, µ2, 1 − µ2) is the best reaction fir Player 3 and a single-proposal equilibrium =⇒ This requires the following rank matrix:

A =



 a₁₁ a₁₂ a₁₃

1 0 0

0 µ₂ 1 − µ₂



 −→



 2 3 2 3 1 1 1 2^∗ 3



 (103)

D₁ = (0.01 0.86 0.03) D^B₂² = (1.00 0.00 0.00) D₃ = (0.00 0.43 0.57)

This can be satisfied if the constraints of D^µ₃ hold. The feasible set S_B7 is then determined by

1. µ₂ < µ₁ =⇒ a₁₂ < 1 + a₁₁ 2. 2a₁₁ < a₂₁ =⇒ a₁₁ < ¹₂ 3. a₂₃ ≤ a₁₃ =⇒ a₁₂ ≤ 1 − a₁₁ 4. 1 − µ₂+ a₂₃ > 2a₁₃ =⇒ a₁₂ > ²₃ − ⁴₃a₁₁ 5. µ₂ > ³₈ =⇒ a₁₂ > ³₄

After comparison of the inequalities only 5. remains.

=⇒ SB7 = ©£

0,¹₄¤

ס₃

4, 1 − a11

¤ª (104)

(See Figure 2)

Minimum payoff for Player 2:

π₂^min(B₇) > 3

8 (105)

Thus we have shown that ∀ D₁ ∈ A ∪ B ∃ a proposal D₂^s of Player 2, such that ρ is symmetric and π₂ > ³₈.

7.2.2 Proof of Corollary 3

1. Suppose D₁ ∈ A₁∪ A₂ ∪ B₁ ∪ B₃∪ B₅ and D₂ is such that ρ is non-symmetric and the best reaction for Player 2, then π₂ ≥ ¹₂, because if D₂ is such that ρ is symmetric Player 2 obtains at least π₂ = ¹₂ (see proof of Proposition 3). Further, we have π₃ > ¹₄ from the proof of Proposition 2. Together with the resource constraint we obtain 1 = π₁+ π₂+ π₃ > ¹₄ +¹₂ + ¹₄ = 1 ¢¢¡¡¢

¢®

2. Suppose D₁ ∈ (A ∪ B)\ {A₁∪ A₂∪ B₁∪ B₃∪ B₅} and D₂ is such that ρ is non-symmetric and π₁ ≥ ¹₄.

First note that π₁ ≥ ¹₄ is only possible if the constraints for D^o₃ or D₃^L² hold, because otherwise Proposition 2 implies π1 = 0 and π2 ≥ ³₈, since the minimum payoff with D₂ such that ρ is symmetric is π₂ = ³₈.

(a) Suppose D₂ is such that the constraints of D₃^o hold =⇒ a =

µ x₁ x₂ y₁ y₂

¶

=⇒ a₂₁+ a₂₂≤ x and a₂₁ = π₁ ≥ ¹₄, a₂₂ = π₂ ≥ ³₈ =⇒ ⁵₈ ≤ x ≤ ¹₂ ¢¢¡¡¢

¢®

(b) Suppose D₂ is such that the constraints of D^L₃^² hold =⇒ a =

µ x₁ x₂ y₁ y₂

¶

and D3 = (0, 2a12−a22+², 1−(2a12−a22+²)) is the best reaction for Player 3 and π1 = ¹₃(a11+ a21) with π2 = ¹₃(2a12− a22+ ² + a12+ a22) = a12+₃^².

i. Suppose D₂ ∈ A\(A₁∪ A₂∪ A₄) =⇒ a₁₂≤ ¹₄ =⇒

π₂ = ¹₄ + ^²₃ < ³₈ (² < ³₈)¢¢¡¡¢¢®

ii. Suppose D₂ ∈ A₄ =⇒ π₁ = ¹₆ ¢¢¡¡¢

¢®

iii. Suppose D2 ∈ B\(B1∪ B3∪ B5) =⇒ a11 ≤ ³₈ and 2a21 < a11 (constraints of D₃^L²) =⇒ π1 < ¹₃(³₈ +₁₆³) = ₁₆³ ¢¢¡¡¢

¢®

(c) Suppose D₂ is such that the constraints of D^L₃^² hold =⇒ a =

µ x₁ x₂ y₁ y₂

¶

and D₃ = (2a₁₁− a₂₁+ ², 0, 1 − (2a₁₁− a₂₁+ ²)) with π₂ = ¹₃(a₁₂+ a₂₂) i. Suppose D₂ ∈ A\(A₁∪ A₂∪ A₄) =⇒ a₁₂≤ ¹₆ and π₂ ≤ ¹₉

since a₂₂≤ a₁₂. ¢¢¡¡¢

¢®

ii. Suppose D2 ∈ A4 =⇒ π2 ≤ ¹₃ < π2(A4) = ¹₂ − ² ¢¢¡¡¢

¢®

iii. Suppose D₂ ∈ B\(B₁∪ B₃∪ B₅) =⇒ a₁₂ ≤ ³₄ and π₂ < ³₈ since 2a₂₂< a₁₂.

7.2.3 Proof of Proposition 4 and Corollary 4 Suppose D₁ ∈ C

C₁) a₁₁+ a₁₂ ≤ ¹₂ and D₂ = (1 − a₂₂, a₂₂, 0) with 1 − (2a₂₂− a₁₂+ ²) = ¹₂(1 − a₁₁− a₁₂) =⇒ a₂₂ = ¹₄(1+3a₁₂+a₁₁−2²) and D₃ = (0, 2a₂₂−a₁₂+²⁰, 1−(2a₂₂−a₁₂+²⁰)) is the best reaction for Player 3 and a single-proposal equilibrium. This requires the following rank matrix:

A =



 a₁₁ a₁₂ a₁₃

1 − a22 a22 0

0 2a₂₂− a₁₂+ ²⁰ 1 − (2a₂₂− a₁₂+ ²⁰)



 −→



 2 1 3 3 2 1 1 3 2^∗



 (106)

D₁ = (0.10 0.10 0.8)

D^C₂¹ = (0.65 + ₂^² 0.35 − ^²₂ 0.0 D₃ = (0.00 0.6 + ² + ²⁰ 0.4 − (² + ²⁰))

This can be satisfied if the constraint set (i) of D³₃ hold with a₂₂= x. The feasible set S_C1 is then determined by

1. a₁₁+ a₁₂ ≤ a₂₂ =⇒ a₁₂ < 1 − 3a₁₁

2. 2a₁₁ < a₂₁ =⇒ a₁₂ ≤ 1 − 3a₁₁

3. 2a12 < a22 =⇒ a12 < ¹₅ +¹₅a11

4. 2a₂₂− a₁₂ < 1 =⇒ a₁₂ ≤ 1 − a₁₁ 5. 1 − (2a₂₂− a₁₂+ ²⁰) > ¹₂(a₁₃+ a₂₃) =⇒ ²⁰ < ²

6. a22 ≤ a21 =⇒ a12 ≤ ¹₃ −¹₃a11

7. 2a₂₁− a₁₁ > 2a₂₂− a₁₂ =⇒ a₁₂ ≤ ¹₂ − a₁₁ After comparison of the inequalities, only 1. and 3. remain.

=⇒ S_C1 = ©¡

0,¹₄¤

×£

0,¹₅ + ¹₅a₁₁¢

∪£₁

4,¹₃¢

× [0, 1 − 3a₁₁)ª

(107) (See Figure 3)

Minimum payoff for Player 2:

π₂^min(C₁) > 1

2 (108)

C₂) D₂ = (1−a₂₂, a₂₂, 0) with 2a₂₁−a₁₁ = 1 =⇒ a₂₁ = ¹₂(1+a₁₁) and a₂₂= ¹₂(1−a₁₁).

D₃ = (0, 2a₂₂− a₁₂+ ², 1 − (2a₂₂− a₁₂+ ²)) is the best reaction for Player 3 and a single-proposal equilibrium. This requires the following rank matrix:

A = feasible set S_C2 is then determined by

1. a₁₁+ a₁₂ ≤ a₂₂ =⇒ a₁₂ ≤ ¹₂ −³₂a₁₁

After comparison of the inequalities, only 1., 3. and 6. remain.

=⇒ S_C2 = ©£

Minimum payoff for Player 2:

π₂^min(C₂) > 1

2 (111)

C₃) D₂ = (1 − a₂₂, a₂₂, 0) with 2a₂₂ − a₁₂ = ¹₂ =⇒ a₂₁ = ¹₄(3 − 2a₁₂) and a₂₂ =

1

4(1 + 2a₁₂). D₃ = (0, 2a₂₂ − a₁₂+ ², 1 − (2a₂₂ − a₁₂+ ²)) is the best reaction for Player 3 and a single-proposal equilibrium. This requires the following rank matrix:

This can be satisfied if the constraint set (ii) of D³₃ holds with a₂₂ = x. The feasible set S_C3 is then determined by

1. a₁₁+ a₁₂ ≤ a₂₂ =⇒ a₁₂ ≤ ¹₂ − 2a₁₁ 2. 2a₁₁ < a₂₁ =⇒ a₁₂ < ³₂ − 4a₁₁

3. 2a₁₂ ≥ a₂₂ =⇒ a₁₂ ≥ ¹₆

4. 2a₂₂− a₁₂ < 1 =⇒ ¹₂ < 1 5. 2a₂₁− a₁₁ ≥ 1 =⇒ a₁₂ ≤ ¹₂ − a₁₁ 6. 1 − (2a₂₂− a₁₂+ ²) ≥ ¹₂(a₁₃+ a₂₃) =⇒ a₁₂ > −a₁₁

7. a₂₂ ≤ a₂₁ =⇒ a₁₂ ≤ ¹₂

After comparison of the inequalities, only 1. and 3. remain.

=⇒ SC3 = ©£

0,¹₆¤

×£₁

6,¹₂ − 2a11

¤ª (113)

(See Figure 3)

Minimum payoff for Player 2:

π₂^min(C₃) > 1

2 (114)

C₄) D₂ = (1 − a₂₂, a₂₂, 0) with 2a₂₂− a₁₂+ ² = 2a₂₁− a₁₁=⇒ a₂₁= ¹₄(2 + a₁₁− a₂₁+ ²) and a21 = ¹₄(2 − a11+ a21− ²). D3 = (0, 2a22− a12 + ²⁰, 1 − (2a22 − a12+ ²⁰)) is the reaction for Player 3 and a single-proposal equilibrium. This requires the following rank matrix:

A =



 a11 a12 a13

1 − a₂₂ a₂₂ 0

0 2a₂₂− a₁₂+ ²⁰ 1 − (2a₂₂− a₁₂+ ²⁰)



 −→



 2 1 3 3 2 1 1 3 2



 (115)

D₁ = (0.06 0.10 0.84)

D₂^C4 = (0.49 + ² 0.51 − ² 0.0 D3 = (0.00 0.92 − ^²₂ + ²⁰ 0.08 +₂^² − ²⁰)

This can be satisfied if the constraint set (i) of D₃^T² holds with a₂₁ = x. The feasible set S_C4 is then determined by

1. a₁₁+ a₁₂ ≤ a₂₁ =⇒ a₁₂ ≤ ²₅ −³₅a₁₁

2. 2a11 < a21 =⇒ a12 ≤ 2 − 7a11

3. 2a₁₂ < a₂₂ =⇒ a₁₂ < ²₇ −¹₇a₁₁ 4. 2a₂₂− a₁₂ < 1 =⇒ a₁₂ ≥ −a₁₁ 5. 1 − (2a22− a12+ ²⁰) < ¹₂(a13+ a23) =⇒ a12 < ¹₂ − a11

6. a₂₁ ≤ a₂₂ =⇒ a₁₂ > a₁₁

7. 2a₂₁− a₁₁ > 2a₂₂− a₁₂ =⇒ ² > 0 After comparison of the inequalities, only 3. and 6. remain.

=⇒ SC4 = ©£

0,¹₄¤

ס

a11,²₇ − ¹₇a11

¢ª (116)

(See Figure 3)

Minimum payoff for Player 2:

π^min₂ (C₄) > 1

2 − ² (117)

C5) In the remaining part SC5 =© C \ S₄

k=1SCk

ª (see Figure 3), we show that, given (a11, a12) ∈ SC5 the share of Player 1 is π1 = 0 if Player 2’s best proposal implies ρ to be non-symmetric or the share of Player 2 is π2 > ³₈ if Player 2’s best proposal implies ρ to be symmetric.

In order to prove this, we divide S_C5 into three subsets:

S_C¹₅ =©

(a₁₁, a₁₂) | a₁₁+ a₁₂ > ¹₂, a₁₁ < ¹₂, a₁₂ < ¹₂ª S_C²₅ =©

(a11, a12) | a11+ a12 ≤ ¹₂, a12≥ ²₇ − ¹₇a11, a12 > ¹₂ − ³₂a11

ª S_C³₅ =©

(a₁₁, a₁₂) | a₁₁+ a₁₂ ≤ ¹₂, a₁₂≥ 1 − 3a₁₁, a₁₁ ≥ 0ª (118) (a) Given (a₁₁, a₁₂) ∈ S_C¹₅ =⇒ with D₂ = (¹₂+ ²,¹₂− ², 0), the inequalities of D^x₃^² hold and D₃ = (0,¹₂− ² + ²⁰,¹₂+ ² − ²⁰) is the best proposal for Player 3 with D₃ being a proposal equilibrium including π₁ = 0 and π₂ = ¹₂ − ² + ²⁰.

i. Suppose Player 2 constructs his proposal D2 in such a way that ρ is non-symmetric and π₁ > 0, then the inequalities of D^L₃^² or D^o₃ must hold (a₁₁= y₁, a₁₂= y₂). But y₁+ y₂ ≤ x cannot hold since x ≤ ¹₂. ii. Suppose Player 2 constructs his proposal D₂ in such a way that ρ is

symmetric and π₂ ≤ ³₈, then this cannot be his best proposal as propos-ing D2 = (¹₂ + ²,¹₂ − ², 0) with non-symmetric ρ and ² < ¹₈ ensures him a share of π₂ > ³₈.

(b) Given (a11, a12) ∈ S_C²₅∪S_C³₅ =⇒ with D2 = (1−(a11+a12−²), a11+a12−², 0) the inequalities of D^x₃^² hold and D₃ = (0, a₁₁+ a₁₂− ² + ²⁰, 1 − (a₁₁+ a₁₂−

² + ²⁰) is the best proposal for Player 3 with D₃ being a proposal equilibrium including π₁ = 0 and π₂ = a₁₂+ a₁₁− ² + ²⁰.

i. Suppose Player 2 constructs his proposal D₂ in such a way that ρ is non-symmetric, π₁ > 0, and the inequalities of D^o₃ hold, with π₂ = a₁₂. This cannot be the best reaction for Player 2 as D₂ = (1−(a₁₁+a₁₂−²), a₁₁+ a₁₂−², 0) with ² < a₁₁ensures him a share of π₂ = a₁₁+a₁₂−²+²⁰ > a₁₂. ii. Suppose Player 2 constructs his proposal D₂ in such a way that ρ is non-symmetric, π₁ > 0 and the inequalities of D₃^L² hold with D₃ = (2a₂₁−a₁₁+², 0, 1−(2a₂₁−a₂₂+²)) is the best proposal for Player 3 (a₁₁ = y₁, a₁₂ = y₂) =⇒ This cannot be the best reaction for Player 2 because his share π₂ = ¹₃(x₂ + y₂) must exceed his share π₂ = y₁ + y₂ + ² − ²⁰ generated by D₂ = (0,¹₂ − ² + ²⁰,¹₂ + ² − ²⁰). But

1

3(x₂+ y₂) ≥ y₁ + y₂ ^{y2≥1−3y1≥}

1 2−³₂y1

=⇒ x₂ = 1^x1=⇒ x^≤1−x2 ₁ = 0 = x ¢¢¡¡¢¢®

(119) since a₁₂≥ ¹₄ and a₁₁+ a₁₂≤ x

iii. Suppose Player 2 constructs his proposal D₂ in such a way that ρ is symmetric and π₂ ≤ ³₈ =⇒

• a₁₁+ a₁₂≤ ³₈ and

• D3 ∈ {D^cor₃ , D₃^L} must hold.

Otherwise Player 2 can obtain a share of π₂ = a₁₁ + a₁₂ − ² + ²⁰ >

3

8 ≥ a₁₁ > a₁₂ by choosing ² < a₁₁ + a₁₂ − ³₈ and proposing D₂ = (1−(a₁₁+a₁₂−²), a₁₁+a₁₂−², 0) with non-symmetric ρ. This also implies D₃ ∈ {D^cor₃ , D₃^L} following from Corollary 5, because if (a₁₁, a₁₂) ∈ {(S_C²₅∪ S_C³₅) ∩ [0,³₈] × [0,³₈− a₁₁]} = {[₁₆⁵,¹₃] × [1 − 3a₁₁,³₈− a₁₁] ∪ [¹₃,³₈] × [0,³₈−a₁₁]} and given symmetric ρ, we have x ≤ a₁₁ ≤ ³₈ ∨ x ≤ a₁₂≤ ₁₆¹. α. Suppose Player 2 constructs his proposal D₂ in such a way that ρ is symmetric and D₃ = D₃^cor is the best proposal for Player 3 leading to a correlated equilibrium C₂₃ =⇒

This cannot be the best reaction for Player 2 because his share π₂(D^cor₃ ) = ¹₂(µ₂+ x₂) = ¹₂(¹₂(a₂₂+ a₁₂) + a₂₂) = ³₄a₂₂+¹₄a₁₂ must exceed π₂(D₃^x²) = a₁₁+ a₁₂− ² + ²⁰, which implies

3

4a₂₂+¹₄a₁₂ ≥ a₁₁+ a₁₂ =⇒ a₂₂ ≥ ⁴₃a₁₁+ a₁₂

a11≥5

a12≥016

≥ ₁₂⁵

(120)

, But this is a contradiction of R(Dˆ ^cor₃ ) =



 3 1 1 2 3 2^∗ 1 2^∗ 3



 (121)

which implies

a₂₃^a11+a12≤

3

≥ 8 ⁵₈ =⇒ a₂₂≤ ³₈ (122) β. Suppose Player 2 constructs his proposal D₂ in such a way that ρ is symmetric and D₃ = D₃^cor is the best proposal for Player 3 leading to a correlated equilibrium C₁₃ =⇒

This cannot be the best reaction for Player 2 since his share is given by π₂ = ¹₂(a₁₂+ a₃₂) ≤ a₁₂ because Φ(a₃₂) ≤ 2.

γ. Suppose Player 2 constructs his proposal D₂ in such a way that ρ is symmetric and D₃ = D₃^Lis the best proposal for Player 3 leading to a selection of the proposal by drawing lots =⇒

This cannot be the best reaction for Player 2 because if a₃₂ = 2a₁₂− a₂₂+ ² we have π₂(D₃^L) = ¹₃(a₁₂+ a₂₂+ a₃₂) = a₁₂+ ^²₃ < π₂(D^x₃^²) and if a₃₂= 0, π₂(D₃^L) > π₂(D^x₃^²) would imply

1

3(a₁₂+ a₂₂) ≥ a₁₁+ a₁₂ =⇒ a₂₂≥ 3a₁₁+ 2a₁₂

a11≥0 a12≥ 1−3a11

a12≤³₈−a11

=⇒ a₂₂≥ 1

(123)

But then we have a₂₃= 0, a₁₃≥ ⁵₈, and a₃₃ ≤ 1, which cannot lead

Thus we have proved Proposition 4 and Corollary 4.

7.2.4 Proof of Proposition 5

In the following, we only give binding constraints.

If a = with D₃ being a single-proposal equilibrium =⇒ π₂ = 0

The solution set is given by Player 3 with D^µ₃ being a single-proposal equilibrium. =⇒ π2 = 0

The solution set is given by S2 = { £ Player 3 with D^x₃ being a single-proposal equilibrium. =⇒ π₂ = 0

The solution set is given by S₃ = ©¡

The solution set is given by S₄ = ©¡

5. Suppose a =

µ 0 1

a21 a22

¶ and

1. a₂₂ ≥ ¹₂

2. µ₁+ 2a₂₂− 1 < a₂₁ =⇒ a₂₂ < ¹₂ +¹₄a₂₁ 3. 1 − µ₁− (2a₂₂− 1) > 2a₂₃ =⇒ a₂₁ > 0

(134)

then the constraints (i) of D^µ₃ⁱhold and D₃^µi = (µ₁, 2a₂₂−1+², 1−(µ₁+2a₂₂−1+²)) is the best reaction for Player 3 with D^µ₃ⁱ being a single-proposal equilibrium. =⇒

π₂ = 2a₂₂− 1 + ² < ¹₅ + ² The solution set is given by

S₅ = ©¡

0,²₅¤

×£₁

2,¹₂ + ¹₄a₂₁¢

∪¡₂

5,¹₂¤

×£₁

2, 1 − a₂₁¤ª

(135) (See Figure 4)

6. Suppose a =

µ 0 1

a₂₁ a₂₂

¶ and

1. a₂₁ > 0

2. µ1+ 2a22− 1 ≥ a21 =⇒ a22 ≥ ¹₂ + ¹₄a21 (136) then the constraints (i) of D₃^x hold and D₃^x= (x, 0, 1 − x) is the best reaction for Player 3 with D^x₃ being a single proposal equilibrium. =⇒ π₂ = 0

The solution set is given by S6 = ©¡

0,²₅¤

×£₁

2 +¹₄a21, 1 − a21

¤ª (137)

(See Figure 4)

Altogether we have shown that D2 = (1, 0, 0) is the best reaction for Player 2 given D1 = (0, 1, 0)

8 Literature

Aumann, R., 1974, Subjectivity and Correlation in Randomized Strategies, Journal of Mathematical Economics, 1, 67-96.

Baron, D.P., Ferejohn, J., 1989, Bargaining in legislatures, American Political Science Review, 83, 1181-1206.

Bernholz, P., Breyer, F., 1994, Grundlagen der Politischen ¨Okonomie, Vol. 2:

Okonomische Theorie der Politik, 3¨ ^rd edition, T¨ubingen.

Binmore, K., Rubinstein, A., Wolinsky, A. 1986, The Nash bargaining solution in economic modelling, Rand Journal of Economics 17, 176-188.

Fudenberg, D., Tirole, J., 1992, Game Theory, The MIT Press, Cambridge, Mas-sachusetts.

Gersbach, H., Wehrspohn, U., 2001, “Dividing a cake by majority decisions”, mimeo.

Harrington, J.E., Jr. 1986, A noncooperative bargaining game with risk averse players and an uncertain finite horizon, Economics Letter 20, 9-13.

Kihlstrom, R., Roth, A.E., Schmeidler, D. , 1981, Risk aversion and solutions to Nash’s bargaining problem, in: O. Moeschlin and D. Pallaschke, eds., Game Theory and Mathematical Economics, North-Holland, Amsterdam.

Krehbiel, K., 1991, Information and Legislative Organization, The University of Michigan Press, Ann Arbor.

Mueller, D.C., 1978, Voting by veto, Journal of Public Economics 10, 57-75.

Myerson, R.B., 1991, Game Theory: Analysis of Conflict, Harvard University Press.

Nielsen, L.T., 1984, Risk sensitivity in bargaining with more than two participants, Journal of Economic Theory 32, 371-376.

Osborne, M.J., 1985, The role of risk aversion in a simple bargaining model, in A.E. Roth, ed., Game-theoretic models of bargaining. Cambridge University Press, Cambridge.

Roth, A.E., 1979, Axiomatic models of bargaining, Lecture Notes in Economics and Mathematical Systems 170, Springer, Berlin.

Roth, A.E., 1985, A note on risk aversion in a perfect equilibrium model of bargaining, Econometrica 53, 207-211.

Roth, A.E., and U.G. Rothblum, 1982, Risk aversion and Nash’s solution for bargaining games with risky outcomes, Econometrica 50, 639-648.

Selten, R., 1975, Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game theory 4, 25-55.

Zeuthen, F., 1930, Problems of monopoly and economic warfare, Rutledge, London.

a₁₂

a₁₁ Feasible set S_A1

a₁₂

a₁₁ Feasible set S_A2

a₁₂

a₁₁ Feasible set S_A3

a₁₂

a₁₁ Feasible set S_A4

a12

a11

Feasible sets S_A1–S_A4

a₁₂

a₁₁ Feasible set S_B1

a₁₂

a₁₁ Feasible set S_B2

a₁₂

a₁₁ Feasible set S_B3

a₁₂

a₁₁ Feasible set S_B4

a₁₂

a₁₁ Feasible set S_B5

a₁₂

a₁₁ Feasible set S_B6

a12

a11

Feasible set S_B7

a12

a11

Feasible sets S_B1–S_B7

a₁₂

a₁₁ Feasible set S_C1

a₁₂

a₁₁ Feasible set S_C2

a₁₂

a₁₁ Feasible set S_C3

a₁₂

a₁₁ Feasible set S_C4

a12

a11

¢¢

¢®¢

S_C²₅

¡¡

¡ ª

S_C¹₅

¡¡

¡ ª

S_C¹₅

©©

©©

©

¼

S_C³₅

Feasible set S_C5

a12

a11

Feasible set S_C1–S_C5

a₁₂

a₁₁ Feasible set S₁

a₁₂

a₁₁ Feasible set S₂

a₁₂

a₁₁ Feasible set S₃

a₁₂

a₁₁ Feasible set S₄

a₁₂

a₁₁ Feasible set S₅

a₁₂

a₁₁ Feasible sets S₆

a12

a11

Feasible set S₁–S₆

In document Cake Division by Majority Decision (Page 37-59)